4.02 Lesson Check Arithmetic Sequences (7)

23.m∠C = 48°m∠A = 90°

Given that the triangle ABC, m∠B = 42∘, a = 10, and c = 23. Now we have to find the missing side length and angles of △ABC. Step-by-step explanation: The formula to find the missing side length of a triangle is given by the Pythagoras theorem. Hence, using the Pythagoras theorem, we get;b2 = c2 - a2 = 23² - 10²b² = 529b = √529b = 23Length of b is 23.We know that the sum of all three angles of a triangle is equal to 180°. Therefore, using this concept, we can find the remaining angles of the triangle as follows;m∠A + m∠B + m∠C = 180°m∠A + 42° + m∠C = 180°m∠A + m∠C = 138°Also, we know that the sum of the two opposite angles of a triangle is equal to the third angle. Therefore, we can use this concept to find one of the remaining angles.m∠A = 180° - m∠C - m∠Bm∠A = 180° - m∠C - 42°m∠A = 138° - m∠C/ Thus,m∠A + m∠C = 138°And,m∠A = 138° - m∠COn solving these two equations, we get,m∠C = 48°m∠A = 90°Using the Pythagoras theorem we find the missing side length of a triangle as given below;b² = c² - a² = 23² - 10²b² = 529b = √529b = 23The length of the missing side is 23.Therefore, the missing side length and angles of △ABC are:Length of b is 23.m∠C = 48°m∠A = 90°

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Answer:

D

Step-by-step explanation:

To find the total number of paintbrushes in the classroom, we need to multiply the number of paintbrushes on each student's desk by the number of students.

Since there are 5 students and each student has 2 paintbrushes, we can use the multiplication operation to find the total number of paintbrushes:

5 students × 2 paintbrushes per student = 10 paintbrushes

Therefore, the correct option is 4. 5 times 2.

[tex](\stackrel{x_1}{1}~,~\stackrel{y_1}{-3})\qquad (\stackrel{x_2}{3}~,~\stackrel{y_2}{-4}) \\\\\\ \stackrel{slope}{m}\implies \cfrac{\stackrel{\textit{\large rise}} {\stackrel{y_2}{-4}-\stackrel{y1}{(-3)}}}{\underset{\textit{\large run}} {\underset{x_2}{3}-\underset{x_1}{1}}} \implies \cfrac{-4 +3}{2} \implies \cfrac{ -1 }{ 2 } \implies - \cfrac{ 1 }{ 2 }[/tex]

[tex]\begin{array}{|c|ll} \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-\stackrel{y_1}{(-3)}=\stackrel{m}{- \cfrac{ 1 }{ 2 }}(x-\stackrel{x_1}{1}) \implies y +3 = - \cfrac{ 1 }{ 2 } ( x -1) \\\\\\ y+3=- \cfrac{ 1 }{ 2 }x+\cfrac{1}{2}\implies y=- \cfrac{ 1 }{ 2 }x+\cfrac{1}{2}-3\implies {\Large \begin{array}{llll} y=- \cfrac{ 1 }{ 2 }x-\cfrac{5}{2} \end{array}}[/tex]

Answer: First, let's find the slope of the line:

slope = (change in y) / (change in x)

slope = (-4 - (-3)) / (3 - 1)

slope = -1/2

Now, let's choose one of the points, say (1,-3), and use the point-slope formula:

y - y1 = m(x - x1)

where m is the slope and (x1, y1) are the coordinates of the point.

So, substituting in the values we get:

y - (-3) = (-1/2)(x - 1)

Simplifying this equation, we get:

y + 3 = (-1/2)x + 1/2

Subtracting 3 from both sides, we get:

y = (-1/2)x - 5/2

Therefore, the equation of the line that passes through the points (1,-3) and (3,-4) is y = (-1/2)x - 5/2.

Your welcome.

Step-by-step explanation:

Her average score on the four remaining tests is indeed 90, which confirms that her lowest score was 80.

Let's assume Peace's lowest score on the five math tests is x.

According to the problem statement, her average mark on all five tests is 88. This means that the sum of her scores on all five tests is:

5 * 88 = 440

If her lowest score was dropped, then the sum of her scores on the remaining four tests would be:

4 * 90 = 360

We know that the sum of her scores on all five tests is 440, so we can write an equation:

440 - x = 360

Solving for x, we get:

x = 80

Therefore, her lowest mark was 80. We can check this by finding her average score after dropping her lowest score:

(88 + 88 + 88 + 88 + 90) / 5 = 90

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Answer: The unit circle is an essential tool in trigonometry because it helps in understanding and visualizing the relationships between angles and the values of the sine, cosine, and tangent functions.

The unit circle is a circle with a radius of one unit and centered at the origin of a coordinate plane. It is divided into 360 degrees or 2π radians. By placing this circle on the coordinate plane, we can easily determine the sine and cosine values of angles in standard position.

For any given angle θ, the sine value is the y-coordinate of the point where the terminal side of the angle intersects the unit circle, and the cosine value is the x-coordinate of that same point. The tangent function, which is the ratio of sine to cosine, can also be determined using the unit circle.

The unit circle also helps in understanding the periodicity of the sine and cosine functions. Since the circumference of the unit circle is 2π, the sine and cosine functions repeat themselves after every 2π radians or 360 degrees. This periodicity allows for the use of trigonometric identities and formulas to simplify and solve complex trigonometric equations.

In summary, the unit circle is an essential tool in trigonometry as it provides a visual representation of angles and their corresponding sine, cosine, and tangent values, and allows for the use of trigonometric identities and formulas to solve complex problems.

Answer:

The answer to your problem is, 156

First lets find out how many feet or in one yard ?

Well 1 yard = 3 feet.

Now that we know that we can use our knowledge to answer the next question:

If Bill ran 52 yards, how many feet did he ran?

We can do math to solve our answer. :

1 x 52 = 52.

Now we need to multiply 52 x 3.

That is a little bit to much, so lets make the problem easier!

By breaking up the problem \/ to make it more easier.

50 x 3. ( 5 x 3 = 15, just add a zero! ) = 150

3 x 2 = 6.

Next add:

150 + 6 = 156.

Thus the answer to your problem is, 156

Answer:

[tex]a_{n}[/tex] = 3 + 5(n - 1)

Step-by-step explanation:

there is a common difference between consecutive terms , that is

8 - 3 = 13 - 8 = 18 - 13 = 5

this indicates the sequence is arithmetic with explicit formula

[tex]a_{n}[/tex] = a₁ + d(n - 1)

where a₁ is the first term and d the common difference

here a₁ = 3 and d = 5 , then

[tex]a_{n}[/tex] = 3 + 5(n - 1) ← is the explicit formula

The maximum value of p is 107.09 and it is obtained when x = 1.44, y = 31.02, z = 0, and w = 0.

To maximize the objective function p = 6x + 9y + 3.3z + 12w

We can use the simplex method.

First, we need to convert the inequalities into equalities by introducing slack variables:

(a) 1.2x + y + z + w + s1 = 121.5

(b) 2x + y − z − w + s2 = 30

(c) 2x + y + z + 1.2w + s3 = 31.5

We can then write the augmented matrix for the problem:

x y z w s1 s2 s3       b

1.2 1 1 1 1 0 0             121.5

2 1 -1 -1 0 1 0               30

2 1 1 1 0 0 1                 31.5

-6 -9 -3.3 -12 0 0 0  0

We choose the most negative coefficient in the bottom row, which is -12. We then select the pivot element in the column corresponding to this coefficient, which is 121.5 in the first row. We perform row operations to make this pivot element equal to 1 and all other elements in its column equal to 0:

x y z w s1 s2 s3                                            b

1 0.83 0.17 0.33 0.67 -0.50 -0.17            100.67

0 0.17 -1.33 -1.33 -0.83 0.50 -0.17   12.33

0 0.17 0.33 0.33 -0.67 -0.50 0.83 10.17

0 -6.50 -12.90 -12.00 4.00 4.50 1.00 408.00

Next, we choose the most negative coefficient in the bottom row, which is -12.9. We select the pivot element in the column corresponding to this coefficient, which is 0.33 in the third row. We perform row operations to make this pivot element equal to 1 and all other elements in its column equal to 0:

x y z w s1 s2 s3                                           b

1 0.00 0.85 0.40 1.44 -0.54 -0.08  107.09

0 0.00 -1.22 -1.67 -1.00 0.38 0.08 -12.93

0 1.00 2.45 2.33 -2.00 1.50 0.33         31.02

0 0.00 -2.19 -5.60 10.00 10.50 1.83 630.00

The objective function value at this point is p = 107.09.

The solution is x = 1.44, y = 31.02, z = 0, w = 0, and the maximum value of p is 107.09.

Therefore, the maximum value of p is 107.09, when x = 1.44, y = 31.02, z = 0, and w = 0.

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Step-by-step explanation:

Let the coordinates of the left endpoint of the line segment be (x1, y1) and the coordinates of the right endpoint be (x2, y2). Let the point dividing the line segment be (x, y). Then the distance between the left endpoint and (x, y) is mx/(m+n) and the distance between (x, y) and the right endpoint is nx/(m+n).

Using the distance formula, we have:

[tex]distance \: between \: (x1, y1) \: and \: (x, y) = \sqrt{((x - x1)^2 + (y - y1)^2)} = \frac{mx}{(m+n)} [/tex]

[tex]distance \: between \: (x, y) \: and \: (x2, y2) = \sqrt{((x2 - x)^2 + (y2 - y)^2)} = \frac{nx}{(m+n)} [/tex]

Squaring both equations, we get:

[tex](x - x1)^2 + (y - y1)^2 = \frac{mx}{(m+n)^2}[/tex]

[tex](x2 - x)^2 + (y2 - y)^2 = \frac{nx}{(m+n)^2}[/tex]

Expanding the squares, we get:

[tex]x^2 - 2x1x + x1^2 + y^2 - 2y1y + y1^2 = \frac{mx}{(m+n)^2} [/tex]

[tex]x^2 - 2x2x + x2^2 + y^2 - 2y2y + y2^2 = \frac{nx}{(m+n)^2}[/tex]

Rearranging and simplifying, we get:

[tex]x = \frac{(mx2 + nx1)}{(m+n)} \: and \: y = \frac{(my2 + ny1)}{(m+n)}[/tex]

Therefore, the point that divides the line segment into two parts so that the ratio of the lengths is m: n and the point is closer to the left endpoint is:

[tex](x, y) = \frac{(mx2 + nx1)}{(m+n)}, \frac{(my2 + ny1)}{(m+n)}[/tex]

V = π(√(803.84/π))^2(12.25) = 9587.15 ft^3. Hence, following Cavalieri's theory, the solid cylinder has a volume of around 9587.15 ft3.

According to Cavalieri's principle, if two solids have the same height and every plane section passing through one solid along the height has the same area as the corresponding plane section passing through the other solid, the two solids must also have the same volume.

A circle of radius r has the same cross-section as a solid cylinder. A = r2 calculates the circle's area.

Therefore, πr^2 = 803.84 r^2 = 803.84/π\sr = √(803.84/π)

The solid cylinder is 12.25 feet tall.

We may get the volume of the solid cylinder using the volume of a cylinder formula, V = r2h: V = π(√(803.84/π))^2(12.25) = 9587.15 ft^3

Hence, following Cavalieri's theory, the solid cylinder has a volume of around 9587.15 ft3.

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Answer:250

Step-by-step explanation:

m is parallel to AB, and passing through points C.

Therefore, m II AB.

In geometry, a line segment is a part of a line bounded by two distinct endpoints and contains every point of the line between its endpoints. The length of a line segment is given by the Euclidean distance between its endpoints. A closed segment includes two endpoints, an open segment does not include both endpoints; a half-open segment has exactly one end. In geometry, a line segment is usually represented using a line above the two end symbols (like AB).

We use the basic rules of construction to draw lines according to the problem.

Draw a straight line AB and take a point C outside of it. Using a ruler and compass to draw the AB line, follow the steps below:

Therefore, the line l is parallel to the line AB.

Complete Question:

Draw a line, say AB, take a point C outside it. Through C, draw a line parallel to AB using ruler and compasses only.

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Number of seconds = Number of minutes x 60

A minute is a unit of time measurement that is equal to 60 seconds. It is commonly used to measure short periods of time, such as the duration of a conversation or the time it takes to complete a task.

Time measurement is the process of quantifying the duration of an event or the interval between two events. It involves measuring the elapsed time between a start point and an end point, or between two points in time.

To convert any number of minutes into seconds, we can use the formula:

Number of seconds = Number of minutes x 60

This formula works because there are 60 seconds in one minute. So, to convert minutes to seconds, we simply multiply the number of minutes by 60 to get the equivalent number of seconds.

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Answer:

(x+5)^2 + (y+12)^2 = 49

Step-by-step explanation:

(x-h)^2 + (y-k)^2 = r ^2 is the general equation for circles, where r is the radius and h,k are the center's coordinates.

It would take them about 8.4 hours to add the special effects if they all work together.

1/T = 1/T1 + 1/T2 + 1/T3 +... + 1/Tn, where T is the time it takes for all n persons to complete the task and T1, T2,..., Tn are the times it takes for each person to do the task individually, is the formula for addressing issues involving work done by several people. This equation is based on the premise that each person's contribution to the activity is inversely proportionate to the time it takes them to do it. In other words, a person contributes more effort to the entire task the faster they work. This method may be used to determine how long it will take all n persons working together to do the assignment in total.

The work done is calculated by:

1/T = 1/T1 + 1/T2 + 1/T3

Here, we have T1 = 20, T2 = 26 and T3 = 32 hours for work done separately.

Substituting the values:

1/T = 1/20 + 1/26 + 1/32

1/T = (32/640) + (24/640) + (20/640)

1/T = 76/640

T = 640/76

T ≈ 8.4 hours

Therefore, it would take them about 8.4 hours to add the special effects if they all work together.

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Baby lizards can run, climb and in some cases even change color to match their surroundings, while baby snakes are generally limited to crawling and slithering.

Baby lizards have developed legs, claws and a tail to help them navigate their environment, while baby snakes have lost their legs during evolution and have developed a long, slender body to help them move around.

Additionally, some baby lizards are born with a protective membrane around their eggs that allows them to move around freely, while baby snakes are usually born inside an egg that they must break out of before hatching. Overall, baby lizards have more mobility and flexibility than baby snakes, which gives them an advantage in many environments.

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The t-statistic is less than the critical value, we fail to reject the null hypothesis

The hypothesis testing to be conducted here is a two-tailed test for the population mean, using the formula:

H0: μ = 7.6 days

Ha: μ ≠ 7.6 days

The critical value is ± 1.960, with a confidence level of 95%.

The sample mean is 7.5 days and the sample standard deviation is 0.9 days. Using the formula, we can calculate the t-statistic:

t = (7.5 - 7.6) / (0.9 / √250) = -1.105

Since the t-statistic is less than the critical value, we fail to reject the null hypothesis, suggesting that Formulation NS does not significantly speed up the healing process compared to the foregoing treatment.

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Answer:

A

Step-by-step explanation:

The explicit functions are:

Arithmetic sequence with common difference d=3: f(n) = 3n +1

Geometric sequence with common ratio r=2: f(n) = 2^n + 2

Quadratic sequence with leading coefficient a=1, constant term c=2, and linear term b=1: f(n) = n^2 + n + 2

There are infinitely many possible sequences that satisfy the condition f(1)=4, so the function to model the value of the nth term in the sequence will depend on the specific pattern or rule that the sequence follows.

Here are some examples of possible sequences and the corresponding explicit functions:

Arithmetic sequence with common difference d=3: f(n) = 4 + 3(n-1) = 3n + 1

Geometric sequence with common ratio r=2: f(n) = 4 x 2^(n-1) = 2^n+2

Quadratic sequence with leading coefficient a=1, constant term c=2, and linear term b=1: f(n) = n^2 + n + 2

Fibonacci sequence with initial values f(1)=f(2)=1: f(n) = ((1+sqrt(5))/2)^n/sqrt(5) - ((1-sqrt(5))/2)^n/sqrt(5)

Note that these are just examples, and there are many other possible functions that could model a sequence with f(1)=4

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Answer:

Yes, the triangles are similar.

Step-by-step explanation:

The triangles are similar because of the three angle measurements.

1) One of the measurements are labeled, and the other two are implied.

2) The measurement of angles HMG and JMK are equal because they are vertical angles.

3) Lastly, because two of the angles are the same, we know that the last angles, GJK and HGJ are the same because the three angles must have a sum of 180°.

The optimal solution of an objective function does not change when its coefficients are changed within a certain range of values. This range of values for each coefficient depends on the function itself and the value of other coefficients.

For example, if the objective function is 3x + 2y, then the range of values for the coefficient of x (3) can be changed from 2 to 4 without changing the optimal solution. Similarly, the coefficient of y (2) can be changed from 1 to 3 without changing the optimal solution.
Therefore, the ranges of values for each coefficient of the objective function, holding all else constant, can be changed without changing the optimal solution.

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Answer:

m = -1

Step-by-step explanation:

Let's change the equation to y = mx + b

m = the slope

x + y = 9

Subtract x both sides

y = -x + 9

m = -1

So, the slope is -1

The standard error of the distribution of sample proportions is approximately 0.090.

The formula to find the standard error of the distribution of sample proportions is given by:Standard error = √(p(1-p)/n)Where,p = Sample proportionn = Sample sizeGiven that the sample proportion is 0.450 and the sample size is 31, we can substitute the values into the formula to find the standard error.Standard error = √(0.450(1-0.450)/31) ≈ 0.090So, the standard error of the distribution of sample proportions is approximately 0.090.

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Answer: At a greengrocer, two bananas and one apple cost $1.16 .

Than the equation becomes

2x + y = 1.16

one banana and one apple cost 0.71.

Than the equation becomes

x + y = 0.71

Subtracting x + y = 0.71 from 2x + y = 1.16

2x - x + y - y = 1.16 - 0.71

x = 0.45

Put in the equation x + y = 0.71

0.45 + y = 0.71

y = 0.71 - 0.45

y = 0.26

The cost of the one apple is  0.26

Step-by-step explanation:

The percentage decrease in the price of the microwave is 51.82%.

Percentage is used to determine the relative value of a digit as a number out of a hundred. The sign that is used to represent percentages is %. In order to express a value as a percentage, multiply the number by 100. Percentage is a measure of frequency.

Percentage decrease = (change in price / initial price) x 100

Change in price = initial price - new price

$137 - $66 = $71

(71/137) x 100 = 51.82%

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According the given question 0.0939 as a Fraction equals 939/10000.

Fractiοns are referred tο as a whοle's cοnstituent parts in mathematics. A single item οr a cοllectiοn οf items can be the cοmplete. When we really cut a slice οf cake frοm the cοmplete cake, the cοmpοnent represents the percentage οf the cake. The wοrd "fractiοn" cοmes frοm Latin.

To write 0.09390 as a fraction you have to write .09390 as numerator and put 1 as the denominator. Now you multiply numerator and denominator by 10 as long as you get in numerator the whole number.

.09390 = .09390/1 = 0.939/10 = 9.39/100 = 93.9/1000 = 939/10000

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Complete Question:

Express 0.09390.0939 as a fraction.

The graphs of f of x equals cosine of the quantity x over 2 and g of x equals square root of 3 minus cosine of the quantity x over 2 intersect on the interval [0, 360°) at: D. 30° and 330°.

In Mathematics and Geometry, a point of intersection simply refers to the location on a graph where two (2) lines intersect or cross each other, which is primarily represented as an ordered pair containing the point that corresponds to the x-coordinate (x-axis) and y-coordinate (y-axis) on a cartesian coordinate.

In this context, we can logically deduce that the point of intersection of the graphs would be at function f(x) = function g(x):

[tex]cos(\frac{x}{2} )=\sqrt{3} -cos(\frac{x}{2} )\\\\\sqrt{3}=cos(\frac{x}{2} ) + cos(\frac{x}{2} )\\\\\sqrt{3}=2cos(\frac{x}{2} )\\\\cos(\frac{x}{2} )=\frac{\sqrt{3}}{2} \\\\cos(\frac{x}{2} )=\frac{\sqrt{3}}{2}\times \frac{\sqrt{3}}{\sqrt{3}} \\\\cos(\frac{x}{2} )= \frac{3}{2\sqrt{3}}[/tex]

x/2 = 15

x = 15(2)

x = 30

x/2 = 165

x = 2(165)

x = 330.

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The probability that the sample proportion surviving for at least 3 years will be less than 70% is approximately 0.2734 or 27.34%.

Probability is a branch of statistics that deals with the study of random events and their likelihood of occurrence.

First, calculate the standard deviation of the sampling distribution of sample proportions using the formula.

σ = [tex]\sqrt{[(p*q)/n]}[/tex], where p is the population proportion, q = (1 - p), and n is the sample size.

In this case, p = 0.74, q = 0.26, and n = 70

Therefore, σ =  [tex]\sqrt{[(0.74*0.26)/70]}[/tex]  = 0.066

Next, we need to standardize the sample proportion using the formula,

z = (X - p) / σ, where X is the sample mean, p is the population proportion, and σ is the standard deviation of the sampling distribution.

In this case, X = 0.70, p = 0.74, and σ = 0.066

Thus, z = (0.70 - 0.74) / 0.066 = -0.606

Using a standard normal distribution table, we find that the cumulative probability for a z-score of -0.606 is 0.2734.

Therefore, the probability that the sample proportion surviving for at least 3 years will be less than 70% is approximately 0.2734 or 27.34%.

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Let's assume Eric's height before summer is x cm. After summer, he became 5% taller, which means his new height is 1.05x cm. We also know that his new height is 151.2 cm. So we can set up an equation:

1.05x = 151.2

To solve for x, we can divide both sides by 1.05:

x = 151.2 / 1.05

x = 144 cm

Therefore, Eric's height before summer was 144 cm.